Key Geometric Theorems and Triangle Classification

1. Thales' Theorem and Similar Triangles

Thales' Theorem is fundamental in geometry. It states that if a straight line is drawn parallel to one side of a triangle, it divides the other two sides proportionally, creating a smaller triangle that is similar to the original one.

The theorem is valid only for similar triangles. For example, given a triangle ABC, if we trace a segment MN parallel to one side, we obtain a smaller triangle AMN similar to ABC.

Criteria for Triangle Similarity

Two triangles are considered similar if they satisfy any of the following premises:

• Their corresponding sides are proportional.
• They have two corresponding angles equal (which implies all three angles are equal).
• They have one equal angle, and the sides forming that angle are proportional.

If triangle ABC is similar to triangle A'B'C', the proportionality of corresponding sides is expressed mathematically as:

$$ \frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'} $$

Where angles A, B, and C are equal, respectively, to angles A', B', and C'.

2. The Pythagorean Theorem

This theorem is valid only for right triangles. A right triangle is defined by:

• Two sides called catheti (or legs), which form the right angle (90°).
• The hypotenuse, which is the side opposite the right angle and is always the longest side of the triangle.

The Pythagorean Theorem states that the sum of the squares of the lengths of the two catheti is always equal to the square of the length of the hypotenuse. Its mathematical statement is:

$$ a^2 + b^2 = c^2 $$

Where a and b are the lengths of the catheti, and c is the length of the hypotenuse.

3. Altitude and Leg Theorems for Right Triangles

These two theorems—the Altitude Theorem and the Leg (Cathetus) Theorem—are valid only for right triangles when an altitude (h) is drawn to the hypotenuse (c), dividing it into two segments, m and n.

The Altitude Theorem

The square of the altitude drawn to the hypotenuse is equal to the product of the lengths of the two segments into which the hypotenuse is divided:

$$ h^2 = m \cdot n $$

The Leg (Cathetus) Theorem

The square of the length of a leg is equal to the product of the length of the hypotenuse and the length of the projection of that leg onto the hypotenuse:

$$ a^2 = m \cdot c \quad \text{and} \quad b^2 = n \cdot c $$

Note on Geometric Proofs and Demonstrations

4. Classification of Triangles

Classification by Side Lengths

• Equilateral: All three sides are equal.
• Isosceles: Exactly two sides are equal.
• Scalene: No sides are equal.

Classification by Internal Angles

• Acute (Acutangles): All internal angles are less than 90 degrees.
• Right (Rectangles): Has exactly one angle equal to 90°.
• Obtuse (Obtusangles): Has exactly one angle greater than 90 degrees.